# mars.tensor.special.yve#

mars.tensor.special.yve(v, z, **kwargs)[source]#

Exponentially scaled Bessel function of the second kind of real order.

Returns the exponentially scaled Bessel function of the second kind of real order v at complex z:

yve(v, z) = yv(v, z) * exp(-abs(z.imag))

Parameters
• v (array_like) – Order (float).

• z (array_like) – Argument (float or complex).

Returns

Y – Value of the exponentially scaled Bessel function.

Return type

ndarray

Notes

For positive v values, the computation is carried out using the AMOS 1 zbesy routine, which exploits the connection to the Hankel Bessel functions $$H_v^{(1)}$$ and $$H_v^{(2)}$$,

$Y_v(z) = \frac{1}{2\imath} (H_v^{(1)} - H_v^{(2)}).$

For negative v values the formula,

$Y_{-v}(z) = Y_v(z) \cos(\pi v) + J_v(z) \sin(\pi v)$

is used, where $$J_v(z)$$ is the Bessel function of the first kind, computed using the AMOS routine zbesj. Note that the second term is exactly zero for integer v; to improve accuracy the second term is explicitly omitted for v values such that v = floor(v).

References

1

Donald E. Amos, “AMOS, A Portable Package for Bessel Functions of a Complex Argument and Nonnegative Order”, http://netlib.org/amos/